This module offers a brief and generally intuitive introduction to maximum likelihood estimation methods. It is intended as a guide for advanced undergraduates.
The maximum likelihood method
Introduction
The maximum likelihood (ML) method is an alternative to ordinary least squares (OLS) and offers a more general approach to the problem of finding estimators of unknown population parameters. In these notes we present an intuitive introduction to the ML technique. We begin our discussion with a description of continuous random variables.
Continuous random variables
Assume that
is a continuous random variable over the interval
Because of the assumption of continuity we need some special definitions.
Probability density function . Any function
that has the following characteristics is a probability density function (pdf): (1)
and (2)
The probability that
x has a value between
a and
b is given by
Here are two examples of the probability density functions (pdf) of continuous random variables.
Uniform distribution
Let
for
and 0 elsewhere, where
A graph of the pdf for this distribution is shown in Figure 1.
It is easy to see from the graph that
and
Moreover, as shown in Figure 1, the area under the pdf curve between
a and
b is equal to the probability that
x lies between
a and
b ; that is,
The calculation of the mean and variance of this distribution is relatively simple. The population mean is given by
or
The population variance
Quite often, as in the exercises at the end of this module, it is easier to calculate the variance of a distribution using the alternative formula for the variance:
where
is given by
Thus,
or
Because of the simple mathematical form of the uniform pdf, the calculations in Example 1 are relatively straight forward. While the calculations for random variables with a pdf that has a more complicated form are generally more difficult (if algebraically possible), the basic methodology remains the same. Example 2 considers the case of a more complicated pdf.
The normal distribution.
A random variable with a mean of
and a variance of
that has a
normal distribution —that is,
has the pdf
A typical graph of this pdf is given in Figure 2. The area under the curve between values of
x of
a and
b is equal to the probability that
x falls between
a and
b .
Joint distributions of samples and the ml method.
Most of the statistical work that economists use involves the use of a sample of observations. It is usual to assume that the members of the sample are drawn independently of each other. The implication of this assumption is that
the pdf of the joint distribution is equal to the product of the pfd of each observation ; i.e.,
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